Calculating Work Required to Change Object Speed

The problem asks for the amount of work required to increase the speed of an object from an initial velocity to a final velocity. This type of problem can be solved using the Work-Energy Theorem, which directly relates the work done on an object to the change in its kinetic energy.

Understanding the Work-Energy Theorem

The Work-Energy Theorem states that the net work done on an object by external forces is equal to the change in its kinetic energy. Mathematically, this is expressed as:

\( W_{\text{net}} = \Delta KE = KE_{\text{final}} - KE_{\text{initial}} \)

Where:

• \( W_{\text{net}} \) is the net work done on the object.
• \( \Delta KE \) is the change in kinetic energy.
• \( KE_{\text{final}} \) is the final kinetic energy.
• \( KE_{\text{initial}} \) is the initial kinetic energy.

Understanding Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It depends on the object's mass and its speed. The formula for kinetic energy is:

\( KE = \frac{1}{2}mv^2 \)

Where:

• \( m \) is the mass of the object.
• \( v \) is the speed (or magnitude of velocity) of the object.

Step-by-Step Calculation of Work

Let's apply the Work-Energy Theorem to find the work required to change the object's speed.

Step 1: Identify the given values.

• Mass of the object, \( m = 4.0 \text{ kg} \)
• Initial speed, \( v_{\text{initial}} = 5.0 \text{ m/s} \)
• Final speed, \( v_{\text{final}} = 10.0 \text{ m/s} \)

Step 2: Calculate the initial kinetic energy (\( KE_{\text{initial}} \)).

Using the kinetic energy formula:

\( KE_{\text{initial}} = \frac{1}{2}mv_{\text{initial}}^2 \)

Substitute the given values:

\( KE_{\text{initial}} = \frac{1}{2} \times 4.0 \text{ kg} \times (5.0 \text{ m/s})^2 \)

\( KE_{\text{initial}} = \frac{1}{2} \times 4.0 \times 25.0 \text{ J} \)

\( KE_{\text{initial}} = 2.0 \times 25.0 \text{ J} \)

\( KE_{\text{initial}} = 50.0 \text{ J} \)

Step 3: Calculate the final kinetic energy (\( KE_{\text{final}} \)).

Using the kinetic energy formula:

\( KE_{\text{final}} = \frac{1}{2}mv_{\text{final}}^2 \)

Substitute the given values:

\( KE_{\text{final}} = \frac{1}{2} \times 4.0 \text{ kg} \times (10.0 \text{ m/s})^2 \)

\( KE_{\text{final}} = \frac{1}{2} \times 4.0 \times 100.0 \text{ J} \)

\( KE_{\text{final}} = 2.0 \times 100.0 \text{ J} \)

\( KE_{\text{final}} = 200.0 \text{ J} \)

Step 4: Calculate the work done using the Work-Energy Theorem.

The work required is the change in kinetic energy:

\( W = KE_{\text{final}} - KE_{\text{initial}} \)

Substitute the calculated kinetic energies:

\( W = 200.0 \text{ J} - 50.0 \text{ J} \)

\( W = 150.0 \text{ J} \)

The amount of work required to be done on the object to increase its speed from 5.0 m/s to 10 m/s is 150 J.

Revision Table: Work and Energy Concepts

Here is a quick review of key concepts related to this problem:

Additional Information: Applying Work-Energy

The Work-Energy Theorem is a powerful tool in physics because it simplifies many problems that would be more complex to solve using only Newton's laws. Instead of analyzing forces and accelerations over time or distance, you can often just look at the initial and final speeds (and thus kinetic energies) and any non-conservative work done.

For instance, if there were friction or air resistance doing work on the object in addition to the work done by the force increasing its speed, the net work would be the sum of all these works, and this net work would equal the total change in kinetic energy.

In this specific problem, we are asked for "the amount of work required to be done on this object". This implies the net work done by forces that increase its speed, assuming no other forces (like friction) are doing significant work, or we are calculating the work done by the force responsible for the acceleration.